## Help on power series and Probability Generating Functions

Right now, as a computer science student, I am having great difficulty with the following problems. I have solved the first one (I think) but only because I essentially looked up the power series and then worked out a general formula for it.

1. Given that the Taylor series for exp(x), write down the
the series for
(exp(x)-exp(-x))/2
and thus write down the series for
(exp(sqrt(x))-exp(-sqrt(x)))/(2 sqrt(x))

2. Give an expression for the sequence an when the generating function is
x^4 (1+x)^-1

3. Give an expression for the sequence an when the generating function is
(2-7x)(1-7x+12x^2)^-1

My lecture notes are being extremely unhelpful and I am at a complete loss as to how to give a proper answer, and the thought process behind it.